60 2 Convex sets. {x a T x b} {x ã T x b}

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1 60 2 Convex sets Exercises Definition of convexity 21 Let C R n be a convex set, with x 1,, x k C, and let θ 1,, θ k R satisfy θ i 0, θ θ k = 1 Show that θ 1x θ k x k C (The definition of convexity is that this holds for k = 2; you must show it for arbitrary k) Hint Use induction on k 22 Show that a set is convex if and only if its intersection with any line is convex Show that a set is affine if and only if its intersection with any line is affine 23 Midpoint convexity A set C is midpoint convex if whenever two points a, b are in C, the average or midpoint (a + b)/2 is in C Obviously a convex set is midpoint convex It can be proved that under mild conditions midpoint convexity implies convexity As a simple case, prove that if C is closed and midpoint convex, then C is convex 24 Show that the convex hull of a set S is the intersection of all convex sets that contain S (The same method can be used to show that the conic, or affine, or linear hull of a set S is the intersection of all conic sets, or affine sets, or subspaces that contain S) Examples 25 What is the distance between two parallel hyperplanes {x R n a T x = b 1} and {x R n a T x = b 2}? 26 When does one halfspace contain another? Give conditions under which {x a T x b} {x ã T x b} (where a 0, ã 0) Also find the conditions under which the two halfspaces are equal 27 Voronoi description of halfspace Let a and b be distinct points in R n Show that the set of all points that are closer (in Euclidean norm) to a than b, ie, {x x a 2 x b 2}, is a halfspace Describe it explicitly as an inequality of the form c T x d Draw a picture 28 Which of the following sets S are polyhedra? If possible, express S in the form S = {x Ax b, Fx = g} (a) S = {y 1a 1 + y 2a 2 1 y 1 1, 1 y 2 1}, where a 1, a 2 R n (b) S = {x R n x 0, 1 T n x = 1, xiai = b1, n xia2 i = b 2}, where a 1,, a n R and b 1, b 2 R (c) S = {x R n x 0, x T y 1 for all y with y 2 = 1} (d) S = {x R n x 0, x T y 1 for all y with n yi = 1} 29 Voronoi sets and polyhedral decomposition Let x 0,, x K R n Consider the set of points that are closer (in Euclidean norm) to x 0 than the other x i, ie, V = {x R n x x 0 2 x x i 2, i = 1,, K} V is called the Voronoi region around x 0 with respect to x 1,, x K (a) Show that V is a polyhedron Express V in the form V = {x Ax b} (b) Conversely, given a polyhedron P with nonempty interior, show how to find x 0,, x K so that the polyhedron is the Voronoi region of x 0 with respect to x 1,, x K (c) We can also consider the sets V k = {x R n x x k 2 x x i 2, i k} The set V k consists of points in R n for which the closest point in the set {x 0,, x K} is x k

2 Exercises 61 The sets V 0,, V K give a polyhedral decomposition of R n More precisely, the sets V k are polyhedra, K V k=0 k = R n, and int V i int V j = for i j, ie, V i and V j intersect at most along a boundary Suppose that P 1,, P m are polyhedra such that m Pi = Rn, and int P i int P j = for i j Can this polyhedral decomposition of R n be described as the Voronoi regions generated by an appropriate set of points? 210 Solution set of a quadratic inequality Let C R n be the solution set of a quadratic inequality, C = {x R n x T Ax + b T x + c 0}, with A S n, b R n, and c R (a) Show that C is convex if A 0 (b) Show that the intersection of C and the hyperplane defined by g T x + h = 0 (where g 0) is convex if A + λgg T 0 for some λ R Are the converses of these statements true? 211 Hyperbolic sets Show that the hyperbolic set {x R 2 + x 1x 2 1} is convex As a generalization, show that {x R n n + xi 1} is convex Hint If a, b 0 and 0 θ 1, then a θ b 1 θ θa + (1 θ)b; see Which of the following sets are convex? (a) A slab, ie, a set of the form {x R n α a T x β} (b) A rectangle, ie, a set of the form {x R n α i x i β i, i = 1,, n} A rectangle is sometimes called a hyperrectangle when n > 2 (c) A wedge, ie, {x R n a T 1 x b 1, a T 2 x b 2} (d) The set of points closer to a given point than a given set, ie, where S R n {x x x 0 2 x y 2 for all y S} (e) The set of points closer to one set than another, ie, where S, T R n, and {x dist(x, S) dist(x, T)}, dist(x, S) = inf{ x z 2 z S} (f) [HUL93, volume 1, page 93] The set {x x + S 2 S 1}, where S 1, S 2 R n with S 1 convex (g) The set of points whose distance to a does not exceed a fixed fraction θ of the distance to b, ie, the set {x x a 2 θ x b 2} You can assume a b and 0 θ Conic hull of outer products Consider the set of rank-k outer products, defined as {XX T X R n k, rank X = k} Describe its conic hull in simple terms 214 Expanded and restricted sets Let S R n, and let be a norm on R n (a) For a 0 we define S a as {x dist(x, S) a}, where dist(x, S) = inf y S x y We refer to S a as S expanded or extended by a Show that if S is convex, then S a is convex (b) For a 0 we define S a = {x B(x, a) S}, where B(x, a) is the ball (in the norm ), centered at x, with radius a We refer to S a as S shrunk or restricted by a, since S a consists of all points that are at least a distance a from R n \S Show that if S is convex, then S a is convex

3 62 2 Convex sets 215 Some sets of probability distributions Let x be a real-valued random variable with prob(x = a i) = p i, i = 1,, n, where a 1 < a 2 < < a n Of course p R n lies in the standard probability simplex P = {p 1 T p = 1, p 0} Which of the following conditions are convex in p? (That is, for which of the following conditions is the set of p P that satisfy the condition convex?) (a) α E f(x) β, where E f(x) is the expected value of f(x), ie, E f(x) = n pif(ai) (The function f : R R is given) (b) prob(x > α) β (c) E x 3 αe x (d) E x 2 α (e) E x 2 α (f) var(x) α, where var(x) = E(x E x) 2 is the variance of x (g) var(x) α (h) quartile(x) α, where quartile(x) = inf{β prob(x β) 025} (i) quartile(x) α Operations that preserve convexity 216 Show that if S 1 and S 2 are convex sets in R m n, then so is their partial sum S = {(x, y 1 + y 2) x R m, y 1, y 2 R n, (x, y 1) S 1, (x, y 2) S 2} 217 Image of polyhedral sets under perspective function In this problem we study the image of hyperplanes, halfspaces, and polyhedra under the perspective function P(x, t) = x/t, with dom P = R n R ++ For each of the following sets C, give a simple description of P(C) = {v/t (v, t) C, t > 0} (a) The polyhedron C = conv{(v 1, t 1),, (v K, t K)} where v i R n and t i > 0 (b) The hyperplane C = {(v, t) f T v + gt = h} (with f and g not both zero) (c) The halfspace C = {(v, t) f T v + gt h} (with f and g not both zero) (d) The polyhedron C = {(v, t) Fv + gt h} 218 Invertible linear-fractional functions Let f : R n R n be the linear-fractional function Suppose the matrix f(x) = (Ax + b)/(c T x + d), dom f = {x c T x + d > 0} [ ] A b Q = c T d is nonsingular Show that f is invertible and that f 1 is a linear-fractional mapping Give an explicit expression for f 1 and its domain in terms of A, b, c, and d Hint It may be easier to express f 1 in terms of Q 219 Linear-fractional functions and convex sets Let f : R m R n be the linear-fractional function f(x) = (Ax + b)/(c T x + d), dom f = {x c T x + d > 0} In this problem we study the inverse image of a convex set C under f, ie, f 1 (C) = {x dom f f(x) C} For each of the following sets C R n, give a simple description of f 1 (C)

4 Exercises 63 (a) The halfspace C = {y g T y h} (with g 0) (b) The polyhedron C = {y Gy h} (c) The ellipsoid {y y T P 1 y 1} (where P S n ++) (d) The solution set of a linear matrix inequality, C = {y y 1A y na n B}, where A 1,, A n, B S p Separation theorems and supporting hyperplanes 220 Strictly positive solution of linear equations Suppose A R m n, b R m, with b R(A) Show that there exists an x satisfying x 0, Ax = b if and only if there exists no λ with A T λ 0, A T λ 0, b T λ 0 Hint First prove the following fact from linear algebra: c T x = d for all x satisfying Ax = b if and only if there is a vector λ such that c = A T λ, d = b T λ 221 The set of separating hyperplanes Suppose that C and D are disjoint subsets of R n Consider the set of (a, b) R n+1 for which a T x b for all x C, and a T x b for all x D Show that this set is a convex cone (which is the singleton {0} if there is no hyperplane that separates C and D) 222 Finish the proof of the separating hyperplane theorem in 251: Show that a separating hyperplane exists for two disjoint convex sets C and D You can use the result proved in 251, ie, that a separating hyperplane exists when there exist points in the two sets whose distance is equal to the distance between the two sets Hint If C and D are disjoint convex sets, then the set {x y x C, y D} is convex and does not contain the origin 223 Give an example of two closed convex sets that are disjoint but cannot be strictly separated 224 Supporting hyperplanes (a) Express the closed convex set {x R 2 + x 1x 2 1} as an intersection of halfspaces (b) Let C = {x R n x 1}, the l -norm unit ball in R n, and let ˆx be a point in the boundary of C Identify the supporting hyperplanes of C at ˆx explicitly 225 Inner and outer polyhedral approximations Let C R n be a closed convex set, and suppose that x 1,, x K are on the boundary of C Suppose that for each i, a T i (x x i) = 0 defines a supporting hyperplane for C at x i, ie, C {x a T i (x x i) 0} Consider the two polyhedra P inner = conv{x 1,, x K}, P outer = {x a T i (x x i) 0, i = 1,, K} Show that P inner C P outer Draw a picture illustrating this 226 Support function The support function of a set C R n is defined as S C(y) = sup{y T x x C} (We allow S C(y) to take on the value + ) Suppose that C and D are closed convex sets in R n Show that C = D if and only if their support functions are equal 227 Converse supporting hyperplane theorem Suppose the set C is closed, has nonempty interior, and has a supporting hyperplane at every point in its boundary Show that C is convex

5 64 2 Convex sets Convex cones and generalized inequalities 228 Positive semidefinite cone for n = 1, 2, 3 Give an explicit description of the positive semidefinite cone S n +, in terms of the matrix coefficients and ordinary inequalities, for n = 1, 2, 3 To describe a general element of S n, for n = 1, 2, 3, use the notation [ ] x1 x 2 x 3 x 1, [ ] x1 x 2, x 2 x Cones in R 2 Suppose K R 2 is a closed convex cone x 2 x 4 x 5 x 3 x 5 x 6 (a) Give a simple description of K in terms of the polar coordinates of its elements (x = r(cos φ, sin φ) with r 0) (b) Give a simple description of K, and draw a plot illustrating the relation between K and K (c) When is K pointed? (d) When is K proper (hence, defines a generalized inequality)? Draw a plot illustrating what x K y means when K is proper 230 Properties of generalized inequalities Prove the properties of (nonstrict and strict) generalized inequalities listed in Properties of dual cones Let K be the dual cone of a convex cone K, as defined in (219) Prove the following (a) K is indeed a convex cone (b) K 1 K 2 implies K 2 K 1 (c) K is closed (d) The interior of K is given by int K = {y y T x > 0 for all x cl K} (e) If K has nonempty interior then K is pointed (f) K is the closure of K (Hence if K is closed, K = K) (g) If the closure of K is pointed then K has nonempty interior 232 Find the dual cone of {Ax x 0}, where A R m n 233 The monotone nonnegative cone We define the monotone nonnegative cone as K m+ = {x R n x 1 x 2 x n 0} ie, all nonnegative vectors with components sorted in nonincreasing order (a) Show that K m+ is a proper cone (b) Find the dual cone K m+ Hint Use the identity n x iy i = (x 1 x 2)y 1 + (x 2 x 3)(y 1 + y 2) + (x 3 x 4)(y 1 + y 2 + y 3) + + (x n 1 x n)(y y n 1) + x n(y y n) 234 The lexicographic cone and ordering The lexicographic cone is defined as K lex = {0} {x R n x 1 = = x k = 0, x k+1 > 0, for some k, 0 k < n}, ie, all vectors whose first nonzero coefficient (if any) is positive (a) Verify that K lex is a cone, but not a proper cone

6 Exercises 65 (b) We define the lexicographic ordering on R n as follows: x lex y if and only if y x K lex (Since K lex is not a proper cone, the lexicographic ordering is not a generalized inequality) Show that the lexicographic ordering is a linear ordering: for any x, y R n, either x lex y or y lex x Therefore any set of vectors can be sorted with respect to the lexicographic cone, which yields the familiar sorting used in dictionaries (c) Find K lex 235 Copositive matrices A matrix X S n is called copositive if z T Xz 0 for all z 0 Verify that the set of copositive matrices is a proper cone Find its dual cone 236 Euclidean distance matrices Let x 1,, x n R k The matrix D S n defined by D ij = x i x j 2 2 is called a Euclidean distance matrix It satisfies some obvious properties such as D ij = D ji, D ii = 0, D ij 0, and (from the triangle inequality) D 1/2 ik D 1/2 ij + D 1/2 jk We now pose the question: When is a matrix D S n a Euclidean distance matrix (for some points in R k, for some k)? A famous result answers this question: D S n is a Euclidean distance matrix if and only if D ii = 0 and x T Dx 0 for all x with 1 T x = 0 (See 833) Show that the set of Euclidean distance matrices is a convex cone 237 Nonnegative polynomials and Hankel LMIs Let K pol be the set of (coefficients of) nonnegative polynomials of degree 2k on R: K pol = {x R 2k+1 x 1 + x 2t + x 3t x 2k+1 t 2k 0 for all t R} (a) Show that K pol is a proper cone (b) A basic result states that a polynomial of degree 2k is nonnegative on R if and only if it can be expressed as the sum of squares of two polynomials of degree k or less In other words, x K pol if and only if the polynomial can be expressed as p(t) = x 1 + x 2t + x 3t x 2k+1 t 2k p(t) = r(t) 2 + s(t) 2, where r and s are polynomials of degree k Use this result to show that { K pol = x R 2k+1 x i = m+n=i+1 Y mn for some Y S k+1 + In other words, p(t) = x 1 + x 2t + x 3t x 2k+1 t 2k is nonnegative if and only if there exists a matrix Y S k+1 + such that (c) Show that K pol = K han where x 1 = Y 11 x 2 = Y 12 + Y 21 x 3 = Y 13 + Y 22 + Y 31 x 2k+1 = Y k+1,k+1 K han = {z R 2k+1 H(z) 0} }

7 66 2 Convex sets and H(z) = z 1 z 2 z 3 z k z k+1 z 2 z 3 z 4 z k+1 z k+2 z 3 z 4 z 5 z k+2 z k+4 z k z k+1 z k+2 z 2k 1 z 2k z k+1 z k+2 z k+3 z 2k z 2k+1 (This is the Hankel matrix with coefficients z 1,, z 2k+1 ) (d) Let K mom be the conic hull of the set of all vectors of the form (1, t, t 2,, t 2k ), where t R Show that y K mom if and only if y 1 0 and y = y 1(1,Eu,Eu 2,,Eu 2k ) for some random variable u In other words, the elements of K mom are nonnegative multiples of the moment vectors of all possible distributions on R Show that K pol = K mom (e) Combining the results of (c) and (d), conclude that K han = cl K mom As an example illustrating the relation between K mom and K han, take k = 2 and z = (1, 0, 0, 0, 1) Show that z K han, z K mom Find an explicit sequence of points in K mom which converge to z 238 [Roc70, pages 15, 61] Convex cones constructed from sets (a) The barrier cone of a set C is defined as the set of all vectors y such that y T x is bounded above over x C In other words, a nonzero vector y is in the barrier cone if and only if it is the normal vector of a halfspace {x y T x α} that contains C Verify that the barrier cone is a convex cone (with no assumptions on C) (b) The recession cone (also called asymptotic cone) of a set C is defined as the set of all vectors y such that for each x C, x ty C for all t 0 Show that the recession cone of a convex set is a convex cone Show that if C is nonempty, closed, and convex, then the recession cone of C is the dual of the barrier cone (c) The normal cone of a set C at a boundary point x 0 is the set of all vectors y such that y T (x x 0) 0 for all x C (ie, the set of vectors that define a supporting hyperplane to C at x 0) Show that the normal cone is a convex cone (with no assumptions on C) Give a simple description of the normal cone of a polyhedron {x Ax b} at a point in its boundary 239 Separation of cones Let K and K be two convex cones whose interiors are nonempty and disjoint Show that there is a nonzero y such that y K, y K

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